Some mistakes in mathematics made by extremely smart and famous people can eventually lead to interesting developments and theorems, e.g. Poincare's 3d sphere charaterization or the search to prove that Euclid's parallel axiom is really necessary unnecessary.

But I also think there are less famous mistakes worth hearing about. So, here's a question:

What's the most interesting mathematics mistake that you know of?

This question is community wiki, meaning neither the question nor the answers receive points (which are reserved for "hard" questions). So please post as much as you like (indeed please post one answer per post so that others can upvote the ones easier), vote a lot and vote freely.

(should there be a tag 'not-math-related' or similar?)

EDIT: There is a similar question which has been closed as a duplicate to this one, but which also garnered some new answers. It can be found here:

Closed: big-list questions don't need to keep cycling back to the front page, after some point.
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Scott Morrison♦Mar 7 '10 at 6:41

10

doesn't "cycling back to the front page" could also mean that it is still of interest? e.g. this one has been just been edited and therefore got to the front page again. Therefore it gets closed??? I don't get the logic behind that...
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vonjdMar 12 '10 at 18:28

11

Well, cycling well-viewed topics back to the front comes at the cost of pushing newer questions out of immediate visibility faster, so I understand the motivation. On the other hand, as the site grows, we get new perspectives on old questions which, and as vonjd points out, are apparently still of interest. We shouldn't close things just because the site old-timers are tired of seeing them. This discussion is probably on meta somewhere already....
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Cam McLemanMar 12 '10 at 18:40

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I agree with Cam - and in this case additionally: the big-list-tag means it is a big list and it can only become a big-list because many people make it a big list - so to close big-lists because they became big-lists is kind of absurd. Perhaps the underlying mechanism of bringing things to the front page should be changed in the software then. Just closing it is no solution
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vonjdMar 12 '10 at 18:46

36 Answers
36

C.N. Little listing the Perko pair as different knots in 1885 (10161 and 10162). The mistake was found almost a century later, in 1974, by Ken Perko, a NY lawyer (!)
For almost a century, when everyone thought they were different knots, people tried their best to find knot invariants to distinguish them, but of course they failed. But the effort was a major motivation to research covering linkage etc., and was surely tremendously fruitful for knot theory.

Update (2013):
This morning I received a letter from Ken Perko himself, revealing the true history of the Perko pair, which is so much more interesting! Perko writes:

The duplicate knot in tables compiled by Tait-Little [3], Conway [1], and Rolfsen-Bailey-Roth [4], is not just a bookkeeping error. It is a counterexample to an 1899 "Theorem" of C.N. Little (Yale PhD, 1885), accepted as true by P.G. Tait [3], and incorporated by Dehn and Heegaard in their important survey article on "Analysis situs" in the German Encyclopedia of Mathematics [2].

Little's `Theorem' was that any two reduced diagrams of the same knot possess the same writhe (number of overcrossings minus number of undercrossings). The Perko pair have different writhes, and so Little's "Theorem", if true, would prove them to be distinct!

Perko continues:

Yet still, after 40 years, learned scholars do not speak of Little's false theorem, describing instead its decapitated remnants as a Tait Conjecture- and indeed, one subsequently proved correct by Kauffman, Murasugi, and Thislethwaite.

I had no idea! Perko concludes (boldface is my own):

I think they are missing a valuable point. History instructs by reminding the reader not merely of past triumphs, but of terrible mistakes as well.

And the final nail in the coffin is that the image above isn't of the Perko pair!!! It's the `Weisstein pair' $10_{161}$ and mirror $10_{163}$, described by Perko as "those magenta colored, almost matching non-twins that add beauty and confusion to the Perko Pair page of Wolfram Web’s Math World website. In a way, it’s an honor to have my name attached to such a well-crafted likeness of a couple of Bhuddist prayer wheels, but it certainly must be treated with the caution that its color suggests by anyone seriously interested in mathematics."

That's a nice mistake. Do you know how it started -- presumably at some point the knots were separated by a flawed computation of some invariant?
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Ryan BudneyDec 16 '09 at 3:06

5

Little (with Tait and Kirkman) compiled his tables combinatorially. He drew all possible 4-valent graphs with some number of vertices (in this case 10), and resolved 4-valent vertices into crossings in all possible ways. He ended up with 2<sup>10</sup> knots. Then he worked BY HAND to eliminate doubles, by making physical models with string. He failed to bring these two knots to the same position, and concluded that they must be different. It took almost 100 years to find the ambient isotopy which shows that there are the same knot, but the quest to show they are different was fruitful.
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Daniel MoskovichDec 16 '09 at 7:22

2

Did Conway assume they were different as well, or did the mistake persist for other reasons, like an error in computing an invariant?
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Ryan BudneyDec 16 '09 at 7:56

6

Ken Perko attempted to make another edit, by adding the following to the citation of Conway's paper: CONWAY WAS NOT MISLED BY THIS FALSE THEOREM OF C.N.LITTLE. HE FOUND THREE COUNTEREXAMPLES AMONG HIS 11-CROSSING NON-ALTERNATING KNOTS AND CORRECTLY WEEDED OUT THE DUPLICATE KNOT TYPES. Cf. Hoste-Thistlethwaite-Weeks, The first 1,701,936 knots, Math. Intelligencer 20 (1998) FOOTNOTE 8 and Jablan-Radovic-Saxdanovic, Adequacy of link fanilies, Publictiones de L'Institute Mathematique, Nouvelle Serie, Tome 88(102) (2010), 21-52.
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S. Carnahan♦Nov 23 '13 at 5:07

4

(Perko's comment continues...) FOR CONWAY, KNOT THEORY WAS A HIGH SCHOOL HOBBY AND HIS CHECKING AND EXTENSION OF THE NINETEENTH CENTURY TABLES "AN AFTERNOON'S WORK." He just didn't look very closely at the 10-crossing knots.
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S. Carnahan♦Nov 23 '13 at 5:08

An error of Lebesgue. 1905 or so. Take a Borel set in the plane, project it onto a line, the result is a Borel set. Obvious: the projection of an open set is open, and the Borel sets in the plane are the least family containing the open sets, closed under countable unions and countable decreasing intersections.

Once upon a time, a set theorist was writing a paper on inner models, and in it he wrote, "... and we will call such models nice." When he got his manuscript back from the typist (this was back in the pre-LaTeX days of technical typists), he saw that his handwriting had been misread, and the line came out as: "... and we will call such models mice." The name stuck, and to this day if you browse almost any recent volume of the Journal of Symbolic Logic, you will find set theory articles on "mice."

I've heard a version of this story too, but I've also heard that Jensen denied that this was the origin of "mice". I never asked Jensen himself about it, so I don't know what to believe.
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Andreas BlassOct 9 '11 at 23:55

9

You know what will be a great paper title? "Of mice and men"
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Aleks VlasevOct 10 '11 at 6:50

Maybe it's not true, but there's the story of the "Grothendieck prime":

One striking characteristic of Grothendieck's mode of thinking is that it seemed to rely so little on examples. This can be seen in the legend of the so-called "Grothendieck prime". In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. "You mean an actual number?" Grothendieck asked. The other person replies, yes, an actual prime number. Grothendieck suggested, "All right, take 57."

But Grothendieck must have known that 57 is not prime, right? Absolutely not, said David Mumford of Brown University. "He doesn’t think concretely."

But does this qualify as an interesting mistake?
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Todd Trimble♦Nov 7 '13 at 14:17

2

@Todd, yes, in the sense that it is a fun mistake.
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JoëlNov 7 '13 at 14:49

3

@Joël I agree that it's an amusing mistake, but I was reading the question more in terms of mistakes that led to interesting developments (and I think the highest voted answers went with that reading).
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Todd Trimble♦Nov 7 '13 at 15:39

An insignificant mistake, but amusing nonetheless: in Cayley's famous 1854 paper where he defines the concept of an abstract group, as an illustration he proves that there are three groups of order 6 (up to isomorphism). This is because he does not realize that the groups $Z_2\times Z_3$ and $Z_6$ are isomorphic. (See my comment for the correct Cayley reference.)

It took me a while to track down the correct reference. It is page 51 of A. Cayley, Desiderta and suggestions: No. 1. The theory of groups, American J. Math. 1 (1878), 50-52. An interesting related paper is G. A. Miller, Contradictions in the literature of group theory, American Math. Monthly 29 (1922), 319-328.
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Richard StanleyMar 1 '10 at 15:51

Harold Edwards wrote a wonderful account of this history in his paper "The background of Kummer's proof of Fermat's last theorem for regular primes". It doesn't seem to be available online, but the mathsci net review is: ams.org/mathscinet-getitem?mr=57:12066a
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Ben LinowitzJan 6 '10 at 2:19

1

I don't know whether it is appropriate to say "discovery" of ideals. Maybe "recognition of the importance/relevance of ideals"?
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Kevin H. LinApr 5 '10 at 6:14

wait... really? this is serious, i use that a lot... dammnit!
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Sean TilsonMar 31 '10 at 6:35

1

Cf. mathoverflow.net/a/35864/27465 and jlms.oxfordjournals.org/content/73/1/65. A sufficient extra structure in an abelian category for this to hold is: Grothendieck's axioms AB3, AB4* and having a set of generators. In particular, it is true in module categories (and even categories of "almost modules").
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Torsten SchoenebergJan 19 '14 at 0:28

Frege was trying to derive the concept of "number" from more basic concepts, and he tried to axiomatize higher-order logic (essentially, a kind of set theory), but his intuitive-seeming axioms were logically inconsistent. Russell first found the inconsistency, which we now call Russell's Paradox.

Pontryagin made a famous mistake while computing the stable homotopy groups of spheres (specifically, π2) which led to the discovery of the Kervaire invariant. I won't spoil what the mistake was: watch this video of Mike Hopkins' talk (second video on the page), starting about 7 minutes in.

"Augustin Louis Cauchy in 1821 published a faulty proof of the false statement that the pointwise limit of a sequence of continuous functions is always continuous. Joseph Fourier and Niels Henrik Abel found counter examples in the context of Fourier series. Dirichlet then analyzed Cauchy's proof and found the mistake: the notion of pointwise convergence had to be replaced by uniform convergence."

I have always loved that way Abel wrote this (in a footnote): «it appears to me that this theorem suffers exceptions»...
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Mariano Suárez-Alvarez♦Dec 15 '09 at 23:49

24

Some (e.g. A. Robinson) say that this is a mis-interpretation of the situation. When Cauchy says the sequence converges at all points this includes infinitesimals and such things not recognized as real numbers nowadays. Abel's counterexample $\sum (1/n) \sin(nx)$ in fact does not converge at certain points $x$ infinitely close to $0$. We can hardly fault Cauchy if he did not use the notion of real number from Dedekind and Cantor, since that would not come until 50 years later.
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Gerald EdgarDec 16 '09 at 16:40

Poincare defined the fundamental group and the homology groups and proved that H _1 was pi _1 abelianized. So the question came up whether there were other groups pi _n whose abelianization would give the H _n. Cech defined the higher pi _n as a proposed answer and submitted a paper on this. But Alexandroff and Hopf got the paper, proved that the higher pi _n were abelian and thus not the solution, and they persuaded Cech to withdraw the paper. Nevertheless a short note appeared and the pi _n started to be studied anyway...

Supposedly Stefan Bergman attended a course on orthogonal functions while an undergraduate, and misunderstood what he was hearing, believing that the functions were supposed to be analytic. This led him to the Bergman kernel and Hilbert spaces of analytic functions, which has developed into a whole field of study at the junction of complex analysis and operator theory, and also with important ramifications in the more geometric parts of SCV. If the story is true, this was certainly an extremely fruitful mistake!

Steiner's count 7776 of the number of the number of plane conics tangent to 5 general plane conics certainly deserves a mention here. He gave this answer in 1848, and it wasn't fixed until 1864, when Chasles pointed out the error and came up with the correct value of 3264. You can regard this as the first recognition of needing appropriate compactifications in order to do valid calculations in enumerative geometry.

Goodrick's "story from Grad school" is incorrect. According to Ronald Jensen, the set theorist in question, he felt that the concept was important enough that it deserved a name which had not already been used elsewhere in mathematics. And 'mice' was it. (Also, note that 'mice' is a noun, and 'nice' is an adjective --- it would not make sense.)

I have heard 3 versions of the origin of the name. They all originated with Jensen, and were told at a rate of one per decade. Last I checked, he actually does not seem to remember the reason for the name.
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Andres CaicedoOct 26 '10 at 4:52

Samuel I. Krieger made many attempts at significant contributions to the field of mathematics, unfortunately some of his efforts did not pan out.

In 1934, he claimed that the 72-digit composite number 231,584,178,474,632,390,847,141,970,017,375,815,706,593,969,331,281,128,078,915,826,259,279,871 was the largest known prime number.

He also attempted to show that the number 2^256(2^257-1) was perfect, implying that 2^257-1 is a prime number. 2^257-1 is actually a composite number: its smallest prime factor is 535,006,138,814,359.

Finally, he claimed to have a counter example to Fermat's Last Theorem x^n + y^n = z^n using the numbers x = 1324, y = 731 and z = 1961 with an undisclosed n. A reporter supposedly called Krieger to ask how the left and the right hand side could be equal, when the left hand side could only end in a 4 or a 6 plus 1, and the right hand side could only end in 1.

Petrovisky-Landis solution to the second part of Hilbert 16th problem.
They "proved" the existence of a bound for the number of limit cycles
of planar polynomial vector fields of fixed degree. Ilyashenko pointed
out the mistake.

The problem remains wide open but the basic idea of Petrovisky-Landis ( complexify )
lead to the study of holomorphic foliations.

In chapter 3 of What Is Mathematics, Really? (pages 43-45), Prof. Hersh writes:

How is it possible that mistakes occur in mathematics?

René Descartes's Method was so clear, he said, a mistake could only happen by inadvertence. Yet, ... his Géométrie contains conceptual mistakes about three-dimensional space.

Henri Poincaré said it was strange that mistakes happen in mathematics, since mathematics is just sound reasoning, such as anyone in his right mind follows. His explanation was memory lapse—there are only so many things we can keep in mind at once.

Wittgenstein said that mathematics could be characterized as the subject where it's possible to make mistakes. (Actually, it's not just possible, it's inevitable.) The very notion of a mistake presupposes that there is right and wrong independent of what we think, which is what makes mathematics mathematics. We mathematicians make mistakes, even important ones, even in famous papers that have been around for years.

Philip J. Davis displays an imposing collection of errors, with some famous names. His article shows that mistakes aren't uncommon. It shows that mathematical knowledge is fallible, like other knowledge.

...

Some mistakes come from keeping old assumptions in a new context.

Infinite dimensionl space is just like finite dimensional space—except for one or two properties, which are entirely different.

...

Riemann stated and used what he called "Dirichlet's principle" incorrectly [when trying to prove his mapping theorem].

Julius König and David Hilbert each thought he had proven the continuum hypothesis. (Decades later, it was proved undecidable by Kurt Gödel and Paul Cohen.)

Sometimes mathematicians try to give a complete classification of an object of interest. It's a mistake to claim a complete classification while leaving out several cases. That's what happened, first to Descartes, then to Newton, in their attempts to classify cubic curves (Boyer). [cf. this annotation by Peter Shor.]

Is a gap in a proof a mistake? Newton found the speed of a falling stone by dividing 0/0. Berkeley called him to account for bad algebra, but admitted Newton had the right answer... Mistake or not?

...

"The mistakes of a great mathematician are worth more than the correctness of a mediocrity." I've heard those words more than once. Explicating this thought would tell something about the nature of mathematics. For most academic philosopher of mathematics, this remark has nothing to do with mathematics or the philosophy of mathematics. Mathematics for them is indubitable—rigorous deductions from premises. If you made a mistake, your deduction wasn't rigorous, By definition, then, it wasn't mathematics!

So the brilliant, fruitful mistakes of Newton, Euler, and Riemann, weren't mathematics, and needn't be considered by the philosopher of mathematics.

Riemann's incorrect statement of Dirichlet's principle was corrected, implemented, and flowered into the calculus of variations. On the other hand, thousands of correct theorems are published every week. Most lead nowhere.

A famous oversight of Euclid and his students (don't call it a mistake) was neglecting the relation of "between-ness" of points on a line. This relation was used implicitly by Euclid in 300 B.C. It was recognized explicitly by Moritz Pasch over 2,000 years later, in 1882. For two millennia, mathematicians and philosophers accepted reasoning that they later rejected.

Can we be sure that we, unlike our predecessors, are not overlooking big gaps? We can't. Our mathematics can't be certain.

The reference to the said article by Philip J. Davis is:

Fidelity in mathematical discourse: Is one and one really two? Amer. Math. Monthly79 (1972), 252–263.

From that article, I find particularly amusing the following couple of paragraphs from page 262:

There is a book entitled Erreurs de Mathématiciens, published by Maurice Lecat in 1935 in Brussels. This book contains more than 130 pages of errors committed by mathematicians of the first and second rank from antiquity to about 1900.There are parallel columns listing the mathematician, the place where his error occurs, the man who discovers the error, and the place where the error is discussed. For example, J. J. Sylvester committed an error in "On the Relation between the Minor Determinant of Linearly Equivalent Quadratic Factors", Philos. Mag., (1851) pp. 295-305. This error was corrected by H. E. Baker in the Collected Papers of Sylvester, Vol. I, pp. 647-650.

...

A mathematical error of international significance may occur every twenty years or so. By this I mean the conjunction a mathematician of great reputation and a problem of great notoriety. Such a conjunction occurred around 1945 when H. Rademacher thought he had solved the Riemann Hypothesis. There was a report in Time magazine.

Euler conjectured that there were no pairs of orthogonal Latin squares for orders $n \equiv 2 \pmod 4$. Nearly two hundred years later, this was proved false for every $n \equiv 2 \pmod 4$ except $2$ and $6$. Here's the link to Euler's paper. Regardless, Euler's work certainly helped spur research into Latin squares.

Karl Pearson's contributions in the development of statistics are so ubiquitous that most users take his assumptions for granted. One key contribution and mistake of his was to claim that all distributions are parametric. Such models are still predominantly used in social and behavioral sciences, but his insistence led to a lot of interesting and very useful developments in mathematical statistics and its applications by people who published refutations of his work (like R.A. Fisher).

For surfaces of constant mean curvature, it is alleged that Hopf thought that all compact CMC surfaces in $\mathbb{R}^3$ were round spheres. CMC surfaces are what you get if you have a soap film bounding a fixed volume, so after a childhood full of blowing bubbles this is a pretty reasonable thing to think. And it even happens to be mostly true: Hopf proved that immersed CMC spheres are round, and Alexandrov proved with a nice reflection argument that embedded CMC surfaces of any genus must actually be round spheres.

But a bit later, Wente discovered a collection of CMC tori. Ivan Sterling has some nice pictures of these on his website, as does MSRI. There are many very pretty connections between these surfaces and algebraic geometry, so to me they sort of mark the start of the modern "integrable systems" era of CMC research.

I should probably add that nobody actually seems sure if Hopf believed that compact CMC surfaces are spheres, but it makes a good creation story for the subfield!